By I.R. Shafarevich (editor), R. Treger, V.I. Danilov, V.A. Iskovskikh
This EMS quantity involves elements. the 1st half is dedicated to the exposition of the cohomology thought of algebraic forms. the second one half offers with algebraic surfaces. The authors have taken pains to offer the cloth conscientiously and coherently. The publication comprises a number of examples and insights on a variety of topics.This ebook can be immensely valuable to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and comparable fields.The authors are recognized specialists within the box and I.R. Shafarevich can be recognized for being the writer of quantity eleven of the Encyclopaedia.
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Extra info for Algebraic geometry 02 Cohomology of algebraic varieties, Algebraic surfaces
The dimensions satisfy 22 = 7+7+8,46 = 10+10+10+16 and 92 = 14 + 14 + 64 respectively. To prove the result in these cases, we use invariant theory: Let e : 9 ----; aff(g) be an etale affine representation arising from an LSA-structure. Let S be the simply connected semisimple algebraic group with Lie algebra 5. The linear part of e is the differential of a rational representation p : S ----; Aff(V). Thus we may regard V as an algebraic S-variety. If the center of 9 is onedimensional, we know that V is isomorphic to a linear S-variety.
On the other hand, we say A E ~. is anti dominant provided 6:(A) is not a positive integer for each 0: E ~+. Proposition 1 implies that any finite-dimensional (ly, Go n Py)-module V determines a corresponding K-homogeneous algebraic vector bundle over the K-orbit Q = K . y. On the other hand, suppose that a finite-dimensional algebraic representation V of K n Py carries a compatible ly-action. Then, whenever the resulting ly-module has an infinitesimal character, we can apply a certain direct image construction [5, Section 4], analogous to the direct image for V-modules [2, Chapter VI, Section 5], to the sections of the algebraic bundle.
We let O(V) denote the resulting sheaf of holomorphic sections. Our main interest will be to describe the representations obtained on the compactly supported sheaf cohomology groups H%(S, O(V)). It turns out that our methods will deliver their most effective results in case the finite-dimensional (py, Go n Py)-module is irreducible and satisfies a certain negativity condition. In order to specify this negativity condition, we introduce the following notations. y) denote the universal enveloping algebra of Iy and suppose Z(ly) C U(ly) is the center of U(ly).